Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Iterations of ln(mod(ln(x^(1/x))) and number with property Left a[4]n=exp (-(a^(n+1))
I have been playing with iterations of

And of course I tried and

is not very interesting, it converges to 2 values depending on integer iteration number. Here are picture of 200 iterations:

is more interesting. This iteration converges to 4 values for each x, cyclicaly,and their dependance on n is shifting depending the region x is in the interval ]0:1[. The convergence values are :

Here is what happens:



When resolution is increased (step decreased) more and more shifts in phase happen in the region which seems to converge to approximately 0,6529204....



This number has properties:

So its 2nd selfroot is , while its reciprocal 1,531580266.. is 3rd superroot of e.

It has also following properties, at least approximately numerically, so it might be wrong, but interesting:

if we denote it , than:




if this is so, what happens if instead of integer n we take x, so that:


I hope I used [4Left] correctly. So:


If accuracy is not enough, so it is numerically only 3-4 digits, perhaps going for n-th superroot of e would improve situation?

Probably this is old knowledge.


Possibly Related Threads…
Thread Author Replies Views Last Post
  Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 18 1,280 09/18/2022, 09:37 PM
Last Post: tommy1729
  Apropos "fix"point: are the fractional iterations from there "fix" as well? Gottfried 12 846 07/19/2022, 03:18 AM
Last Post: JmsNxn
Question Is the Tetra-Euler Number Rational? Catullus 1 194 07/17/2022, 06:37 AM
Last Post: JmsNxn
  [note dump] Iterations and Actions MphLee 23 4,407 07/15/2022, 04:08 PM
Last Post: MphLee
Question Natural Properties of the Tetra-Euler Number Catullus 6 578 07/01/2022, 08:16 AM
Last Post: Catullus
  Consistency in the composition of iterations Daniel 9 661 06/08/2022, 05:02 AM
Last Post: JmsNxn
  Has anyone solved iterations of z+Γ(z)? Leo.W 5 2,219 01/07/2022, 08:15 AM
Last Post: JmsNxn
  Dynamical Systems and Number Theory Daniel 4 2,307 06/01/2021, 11:34 PM
Last Post: JmsNxn
  [MSE] Shape of orbit of iterations with base b on Shell-Thron-region Gottfried 14 22,110 12/13/2019, 02:33 PM
Last Post: Ember Edison
  Math overflow question on fractional exponential iterations sheldonison 4 10,720 04/01/2018, 03:09 AM
Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)